Question

Evaluate:
${\int }_0^{1}x(1-x{)}^{5}dx$

Answer 1

${\int }_0^{1}x(1-x{)}^{5}dx$

Let $t=1-x\Rightarrow dt=-dx$

when $x=0\Rightarrow t=1,x=1\Rightarrow t=0$

${\int }_0^1(1-t)t^5(-dt)=+{\int }_0^1(t-1)t^{5}dt$

$={\int }_0^1(t^6-t^5)dt$

= $-[\frac{t^7}{7}-\frac{t^6}{6}{]}_0^1$

$=-[\frac{1}{7}-\frac{1}{6}]=-\left[\frac{6-7}{42}\right]$

$=\frac{1}{42}$